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A New Composite Fractal Function and Its Application in Image Encryption

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A New Composite Fractal Function and Its Application in Image Encryption Journal of Imaging Article A New Composite Fractal Function and Its Application in Image Encryption Shafali Agarwal Independent Researcher, 9600 Coit Road, Plano, TX 75025, USA; [email protected]; Tel.: +1-916-693-4645 Received: 28 May 2020; Accepted: 10 July 2020; Published: 15 July 2020 Abstract: Fractal’s spatially nonuniform phenomena and chaotic nature highlight the function utilization in fractal cryptographic applications. This paper proposes a new composite fractal function (CFF) that combines two different Mandelbrot set (MS) functions with one control parameter. The CFF simulation results demonstrate that the given map has high initial value sensitivity, complex structure, wider chaotic region, and more complicated dynamical behavior. By considering the chaotic properties of a fractal, an image encryption algorithm using a fractal-based pixel permutation and substitution is proposed. The process starts by scrambling the plain image pixel positions using the Henon map so that an intruder fails to obtain the original image even after deducing the standard confusion-diffusion process. The permutation phase uses a Z-scanned random fractal matrix to shuffle the scrambled image pixel. Further, two different fractal sequences of complex numbers are generated using the same function i.e., CFF. The complex sequences are thus modified to a double datatype matrix and used to diffuse the scrambled pixels in a row-wise and column-wise manner, separately. Security and performance analysis results confirm the reliability, high-security level, and robustness of the proposed algorithm against various attacks, including brute-force attack, known/chosen-plaintext attack, differential attack, and occlusion attack. Keywords: composite fractal function; henon map; z-scan; random fractal matrix; permutation; diffusion 1. Introduction A new digital world has increased the transmission demand of multimedia content at large scale. At the same time, excessive use of digital media such as social networking sites and instant messaging brings about the risk of invading to the personal privacy of its owner. Sometimes, this is required to be more conscious about the secure receiving of image contents like in military security documents, medical diagnostic images, bank account-related information from financial institutions, government offices, etc. Thus, it is of the utmost concern to protect the system from the serious risk of confidential data leakage [1]. The image encryption technique was introduced to secure the images transmitted through the smartphone, IPAD, computer, laptop, and other electronic devices over the internet or outsourced to the cloud storage. As compared to text and binary data, image encryption is more critical due to its additional characteristics such as highly correlated adjacent pixels in different directions, excessive redundancy, and greater data volume. Although many conventional encryption algorithms such as Data Encryption Standard [2], RSA [3], Advanced Encryption Standard [4], and International Data Encryption Algorithm [5] with high data transmission security have been proposed. However, these ciphers proved inappropriate and time consuming for image information encryption. The foremost concern of implementing image-specific cryptosystem to resolute image information protection includes the consideration of all necessary image characteristics. Over the years, various cryptosystem design J. Imaging 2020, 6, 70; doi:10.3390/jimaging6070070 www.mdpi.com/journal/jimaging J. Imaging 2020, 6, 70 2 of 27 techniques have been developed, comprising a chaotic system, DNA structure, fractal function, wavelet transform, cellular automata [6–11], and many more. The author in [12] suggested that a cryptosystem having two fundamental structures, i.e., confusion-diffusion, exhibits resilient performance against cyber-attacks. The confusion phase deals with the pixel shuffling or pixel permutation to reduce the relationship between the adjacent image pixels. Whereas pixel diffusion ensures that the small change in plain image information should affect at least half of the cipher image information. Pixel shuffling is achieved by changing the pixel positions within the image, and the pixel diffusion is performed by changing the pixel values. Both phases can be repeated as many times as needed to attain the desired security level. However, it costs the total execution time. Over the last decade, many image encryption ideas have been proposed using the same structure [10,13–18]. An image encryption algorithm can perform permutation in two ways: (1). Bit-level permutation [19–21], and (2). Pixel-based permutation [22–24]. A bit is considered as the smallest operating element. To perform a bit-level permutation, an image needs to be transformed into the binary form. A bit-plane scrambling shuffles the bits itself, and hence, is considered more secure. However, it requires more execution time compared to the pixel-plane scrambling. For example, the author in [25] proposed a bit-level permutation by generating a new sequence dynamically on every small change in the plain image to ensure the robustness of the system against the chosen-plaintext and known-plaintext attacks. Shouliang Li et al. pointed out that by implementing bit-plane and pixel-plane scrambling jointly can have better performance in terms of security as well as a speedy outcome [26]. A bit-plane scrambling plus diffusion (SPD) operation was proposed based on the card trick method to change the pixel position and value at the same time [14]. Before performing it, a pixel-level scrambling is also done to randomize the plain image pixels. Another example of simultaneous pixel confusion and diffusion was proposed in [13]. Both processes were implemented through horizontal confusion-diffusion followed by vertical confusion-diffusion with the help of a newly proposed chaotic map, i.e., tent delay-sine cascade with the logistic map. A high-efficient pixel scrambling was implemented in a block-based square matrix. The method used a chaotic sequence to shuffle the pixels among the different blocks, following which the pixel positions were changed again within the block itself using another chaotic sequence [23]. A simple yet difficult to breach cryptosystem was designed by the author [24], in which pixel-based scrambling was implemented using the Arnold method and Lucas series. A total of nine rounds of pixel permutation were executed to enhance the efficiency and uniformity of the pixels. Xuncai Zhang et al. came up with a concept of local pixel scrambling and global pixel scrambling, which was executed multiple times using a Hilbert curve and piecewise linear chaotic map (PWLCM) respectively [22]. The overall encryption process performed by executing multiple rounds of pixel scrambling and pixel diffusion alternatively. A two-pass method of pixel permutation introduces the utilization of the first half-pixel values to permute the other second half and vice-versa [27]. The author proposed a simple and fast one-round-based dynamic diffusion operation in which the sum of the plain image pixels plays an important role to have an extremely sensitive and secure cryptosystem. At each pixel diffusion step, the sum of the pixels will be updated to significantly increase the resistivity against the known-plaintext, chosen-plaintext, and differential attacks [28]. Recently, a two-steps diffusion process was proposed in [29]. In the method, a new two-dimensional chaotic infinite collapse map generates two matrices X and Y to perform pixel substitution. In the first step, scrambled image pixel values were rearranged according to the sorting index generated through a one-dimensional chaotic sequence X. Further, perform the diffusion operation on a previously generated sequence using another chaotic matrix Y. A fractal is a self-similar image that is infinitely scalable in all directions and encompasses striking features to study and explore. The fractal function works in a complex domain and generates complex patterns using complex numbers upon execution. The chaotic nature of the fractal function exhibits extremely random behavior and excellent sensitivity towards its initial condition, which makes eavesdroppers inefficient to replicate the secret key to retrieve the confidential information [30–33]. J. Imaging 2020, 6, 70 3 of 27 Moreover, a complicated structure of a fractal function leads to a reliable cryptosystem design that is capable of secure digital information transmission. A fractal possesses a small footprint, which means only a few parameters are required to produce a unique fractal image. The image instantly reacts to even a tiny changed input, resulting in a completely different fractal image. At an early stage, the USA navy published a patent describing the use of fractal function as a secret key in an image encryption method [34]. However, a fractal function can be used in any phase of a cryptosystem design, i.e., key design or encryption/decryption. The author utilized a superior fractal function to generate three initial values, which were further inputted to a newly proposed chaotic map to produce three secret key sequences in [35]. The Mandelbrot set, a most famous fractal function introduced by B.B. Mandelbrot in 1979 [33], has a wide exploration history due to its interesting and complex structure. In the last decade, many image encryption methods have been proposed using the Mandelbrot set and a Julia set [36–40]. A set of four Julia images was used in the secret key design, which was further modified
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